The number of maximal chains of maximal size in a poset.

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.

The multiplicity of the largest part of an integer partition.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. The $k$-Gorenstein degree is the maximal number $k$ such that the algebra is $k$-Gorenstein. We apply the convention that the value is equal to the global dimension of the algebra in case the $k$-Gorenstein degree is greater than or equal to the global dimension.

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. Associate to this special CNakayama algebra a Dyck path as follows: In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra. The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.